Conservatives accused Mr Johnson of making "clumsy threats" over the issue.

Private schools claim many of them would close if not for the annual £100m in tax breaks from being a charity.

New charity rules mean private schools now have to pass a public benefit test showing how they add to communities.

Some teachers told the BBC that they found it patronising of Mr Johnson to imply that state school teachers needed the help of private school teachers.

So they have these facilities, they should make those facilities open to state school pupils as well

Alan JohnsonEducation Secretary

And a spokeswoman for the National Union of Teachers said it was not "a one-way street".

"Private schools have a massive amount to learn from state schools.

"Teaching 30-35 pupils is very different from teaching 15 and requires a totally different set of skills and is much more demanding. That's the big problem that private school teachers will have," she said.

Specialist teachers

Speaking at a hustings meeting in Bristol, Mr Johnson said the Charity Commission was looking to update what private schools had to do to earn their charitable status.

He said: "It shouldn't just be access to a sports field and the occasional amateur dramatic society open event."

Mr Johnson said private schools tended to get more specialist teachers and spend more money on facilities such as science labs.

He added: "So they have these facilities, they should make those facilities open to state school pupils as well, and they should join in a partnership with state schools in order to get their charitable status. And I think that's crucial."

In an interview with the Daily Telegraph, Mr Johnson outlined his proposal, saying the independent education sector's charitable status meant it needed to contribute more to its surrounding communities.

Schools Minister Lord Adonis said the argument that private schools should justify their charitable status was government policy, and not simply Mr Johnson's idea.

He said: "It's very much a government policy. It follows the Charities Act, that was enacted last year, which requires public benefit to be demonstrated.

"And I should stress that it's not something that the government itself will be undertaking, this is something for the Charity Commission, which is a wholly independent body, whose job is simply to implement the law."

'Two-way street'

Also speaking at the hustings meeting, Mr Cruddas said he had "never understood" the charitable status for private schools.

"I would get rid of it and invest that money in the public sector," he said.

Shadow Education Secretary David Willetts said co-operation between schools should be a "two-way street", with state schools making a contribution.

I think Alan Johnson needs to pick his words carefully because I think it does sound patronising to the wonderful teachers we have in state schools

John O'FarrellSchool governor

"As Alan Johnson is running for the deputy leadership he is making clumsy threats about independent schools' charity status," he said.

"Instead we would be far more positive, encouraging independent schools and maintained schools to come together through creating new academies."

Anthony Seldon, the master of Wellington College in Berkshire, told BBC Radio 4's Today programme that he would be happy to lend his teachers to state schools.

"It's been a great tragedy for British education over the last hundred years that there has been an apartheid between the state and independent sectors - and this is not helpful for the children, nor the teachers, nor the country at large."

He said there were many more highly qualified graduate teachers in the sciences, languages and maths in the private sector, who should be shared in a "non-patronising way" with the state sector.

Mr Seldon added that there were many ways in which private schools already shared facilities and pointed out that parents pay fees as well as taxes.

Chinese teachers

Some teachers told the programme that it was an insult to suggest that state school teachers did not do as good a job as private school ones.

Novelist John O'Farrell, who chairs the governors of Lambeth Academy in South London, agreed, saying that state school teachers had to teach more varied communities than their private school counterparts.

"We don't really need private school teachers in the state system.

"What might be useful is the odd highly specialised teacher, teachers of Chinese or something, but that aside I think Alan Johnson needs to pick his words carefully because I think it does sound patronising to the wonderful teachers we have in state schools," said Mr O'Farrell.

Mr Johnson has submitted a series of proposals to the charities' regulator, the Charities Commission.

Sarah Atkinson from the Charity Commission said schools, like every other charity, must show that they benefit the public to justify their charity status.

"That means that they've got to show a real benefit to the public, so not just a private interest," she said.

"And crucially they've got to show how people on low incomes can benefit as well."